Question

If S and T are two sets such that S has 21 elements, T has 32 elements, and $S \cap T$ has 11 elements, how many elements does $S \cup T$

Answer 1

We have to sets S and T. The number of elements in each set is given and the number of elements in the intersection of both sets is also given. Then we can find the number of elements in the union of both sets using the equation $n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$

Complete step by step solution:
We are given to sets S and T.
The number of elements of S is given as 21
$\Rightarrow n\left( S \right) = 21$
The number of elements of T is given as 32
$\Rightarrow n\left( T \right) = 32$
And number of elements of $S \cap T$ is 11
$\Rightarrow n\left( {S \cap T} \right) = 11$
We need to find the number of elements in $S \cup T$ or $n\left( {S \cup T} \right)$
We know that, $n\left( {S \cup T} \right) = n\left( S \right) + n\left( T \right) - n\left( {S \cap T} \right)$
Substituting the values in the equation, we get,
$n\left( {S \cup T} \right) = 21 + 32 - 11$
On solving we get,
$\Rightarrow n\left( {S \cup T} \right) = 42$

Therefore, the number of elements in $S \cup T$ is 42.

Note:
The concept used here is set theory. Union of two sets gives a set of all elements that are at least in one of the two sets. If A and B are 2 sets, then its union is written as $A \cup B$ and it contains all the elements in A along with all the elements in B. Intersection of two sets gives the set of all elements that are in both the sets. If A and B are 2 sets, then its intersection is written as $A \cap B$ and it contains all the elements that are both in A and B. For a set A, $n\left( A \right)$ represents the number of elements in the set.